Localization and "classical entanglement'' in the Discrete Non-Linear Schrödinger Equation
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| Format: | Preprint |
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2025
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| _version_ | 1866908677974261760 |
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| author | Giachello, Martina Iubini, Stefano Livi, Roberto Gradenigo, Giacomo |
| author_facet | Giachello, Martina Iubini, Stefano Livi, Roberto Gradenigo, Giacomo |
| contents | We perform a detailed numerical study of the very peculiar thermodynamic properties of the localized high-energy phase of the Discrete Non-Linear Schrödinger Equation (DNLSE). A numerical sampling of the microcanonical ensemble done by means of Hamiltonian dynamics reveals a new and subtle relation between the presence of the localized phase and a property of the system that we have called {\it ``classical entanglement''}. Our main finding is that a quantity defined for our classical system in perfect analogy with the entanglement entropy of quantum ones, and that we have therefore called $S_{\mathrm{ent}}$, grows with the system size $N$ in the localized phase as $S_{\mathrm{ent}}(N) \sim \log(N)$, therefore revealing the presence of subtle non-local correlations between any finite portion of the system and the rest of it. This manifestation of {\it ``classical entanglement''} beautifully captures the lack of system separability in the DNLSE localized phase, revealing how statistical correlations specific to the microcanonical ensemble and non-reproducible in the canonical one, may concur to determine a property totally analogous to the one produced by non-local quantum correlations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Localization and "classical entanglement'' in the Discrete Non-Linear Schrödinger Equation Giachello, Martina Iubini, Stefano Livi, Roberto Gradenigo, Giacomo Statistical Mechanics Quantum Physics We perform a detailed numerical study of the very peculiar thermodynamic properties of the localized high-energy phase of the Discrete Non-Linear Schrödinger Equation (DNLSE). A numerical sampling of the microcanonical ensemble done by means of Hamiltonian dynamics reveals a new and subtle relation between the presence of the localized phase and a property of the system that we have called {\it ``classical entanglement''}. Our main finding is that a quantity defined for our classical system in perfect analogy with the entanglement entropy of quantum ones, and that we have therefore called $S_{\mathrm{ent}}$, grows with the system size $N$ in the localized phase as $S_{\mathrm{ent}}(N) \sim \log(N)$, therefore revealing the presence of subtle non-local correlations between any finite portion of the system and the rest of it. This manifestation of {\it ``classical entanglement''} beautifully captures the lack of system separability in the DNLSE localized phase, revealing how statistical correlations specific to the microcanonical ensemble and non-reproducible in the canonical one, may concur to determine a property totally analogous to the one produced by non-local quantum correlations. |
| title | Localization and "classical entanglement'' in the Discrete Non-Linear Schrödinger Equation |
| topic | Statistical Mechanics Quantum Physics |
| url | https://arxiv.org/abs/2503.14364 |